A Property of an Autonomous ODE

I wonder how to prove that if y(t)=sin(t) solves an autonomous ODE f(y,y',...,y^(n))=0, then x(t)=cos(t) is also a solution.

I mean I'm a bit distracted by the fact that all derivatives of y are present here. For example in the equation for a pendulum there are just y and y'' and a special functional dependence between them and that's why it works, isn't it?

In the above case I know almost nothing about the function f. So how to proceed?